Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇐)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇐)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔)
        ↳22 YES

(0) Obligation:

Clauses:

list([]).
list(.(X, XS)) :- list(XS).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), list(XS)).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

s2l9(s(T16), .(X65, X66)) :- s2l9(T16, X66).
s2l9(0, []).
list21([]).
list21(.(T28, T30)) :- list21(T30).
goal1(s(T8)) :- s2l9(T8, X28).
goal1(s(T8)) :- ','(s2l9(T8, T23), list21(T23)).
goal1(0).

Queries:

goal1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
s2l9_in: (b,f)
list21_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X28))
s2l9_in_ga(s(T16), .(X65, X66)) → U1_ga(T16, X65, X66, s2l9_in_ga(T16, X66))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X65, X66, s2l9_out_ga(T16, X66)) → s2l9_out_ga(s(T16), .(X65, X66))
U3_g(T8, s2l9_out_ga(T8, X28)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, list21_in_g(T23))
list21_in_g([]) → list21_out_g([])
list21_in_g(.(T28, T30)) → U2_g(T28, T30, list21_in_g(T30))
U2_g(T28, T30, list21_out_g(T30)) → list21_out_g(.(T28, T30))
U5_g(T8, list21_out_g(T23)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
list21_in_g(x1)  =  list21_in_g(x1)
[]  =  []
list21_out_g(x1)  =  list21_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X28))
s2l9_in_ga(s(T16), .(X65, X66)) → U1_ga(T16, X65, X66, s2l9_in_ga(T16, X66))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X65, X66, s2l9_out_ga(T16, X66)) → s2l9_out_ga(s(T16), .(X65, X66))
U3_g(T8, s2l9_out_ga(T8, X28)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, list21_in_g(T23))
list21_in_g([]) → list21_out_g([])
list21_in_g(.(T28, T30)) → U2_g(T28, T30, list21_in_g(T30))
U2_g(T28, T30, list21_out_g(T30)) → list21_out_g(.(T28, T30))
U5_g(T8, list21_out_g(T23)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
list21_in_g(x1)  =  list21_in_g(x1)
[]  =  []
list21_out_g(x1)  =  list21_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(s(T8)) → U3_G(T8, s2l9_in_ga(T8, X28))
GOAL1_IN_G(s(T8)) → S2L9_IN_GA(T8, X28)
S2L9_IN_GA(s(T16), .(X65, X66)) → U1_GA(T16, X65, X66, s2l9_in_ga(T16, X66))
S2L9_IN_GA(s(T16), .(X65, X66)) → S2L9_IN_GA(T16, X66)
GOAL1_IN_G(s(T8)) → U4_G(T8, s2l9_in_ga(T8, T23))
U4_G(T8, s2l9_out_ga(T8, T23)) → U5_G(T8, list21_in_g(T23))
U4_G(T8, s2l9_out_ga(T8, T23)) → LIST21_IN_G(T23)
LIST21_IN_G(.(T28, T30)) → U2_G(T28, T30, list21_in_g(T30))
LIST21_IN_G(.(T28, T30)) → LIST21_IN_G(T30)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X28))
s2l9_in_ga(s(T16), .(X65, X66)) → U1_ga(T16, X65, X66, s2l9_in_ga(T16, X66))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X65, X66, s2l9_out_ga(T16, X66)) → s2l9_out_ga(s(T16), .(X65, X66))
U3_g(T8, s2l9_out_ga(T8, X28)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, list21_in_g(T23))
list21_in_g([]) → list21_out_g([])
list21_in_g(.(T28, T30)) → U2_g(T28, T30, list21_in_g(T30))
U2_g(T28, T30, list21_out_g(T30)) → list21_out_g(.(T28, T30))
U5_g(T8, list21_out_g(T23)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
list21_in_g(x1)  =  list21_in_g(x1)
[]  =  []
list21_out_g(x1)  =  list21_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U4_G(x1, x2)  =  U4_G(x2)
U5_G(x1, x2)  =  U5_G(x2)
LIST21_IN_G(x1)  =  LIST21_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_G(s(T8)) → U3_G(T8, s2l9_in_ga(T8, X28))
GOAL1_IN_G(s(T8)) → S2L9_IN_GA(T8, X28)
S2L9_IN_GA(s(T16), .(X65, X66)) → U1_GA(T16, X65, X66, s2l9_in_ga(T16, X66))
S2L9_IN_GA(s(T16), .(X65, X66)) → S2L9_IN_GA(T16, X66)
GOAL1_IN_G(s(T8)) → U4_G(T8, s2l9_in_ga(T8, T23))
U4_G(T8, s2l9_out_ga(T8, T23)) → U5_G(T8, list21_in_g(T23))
U4_G(T8, s2l9_out_ga(T8, T23)) → LIST21_IN_G(T23)
LIST21_IN_G(.(T28, T30)) → U2_G(T28, T30, list21_in_g(T30))
LIST21_IN_G(.(T28, T30)) → LIST21_IN_G(T30)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X28))
s2l9_in_ga(s(T16), .(X65, X66)) → U1_ga(T16, X65, X66, s2l9_in_ga(T16, X66))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X65, X66, s2l9_out_ga(T16, X66)) → s2l9_out_ga(s(T16), .(X65, X66))
U3_g(T8, s2l9_out_ga(T8, X28)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, list21_in_g(T23))
list21_in_g([]) → list21_out_g([])
list21_in_g(.(T28, T30)) → U2_g(T28, T30, list21_in_g(T30))
U2_g(T28, T30, list21_out_g(T30)) → list21_out_g(.(T28, T30))
U5_g(T8, list21_out_g(T23)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
list21_in_g(x1)  =  list21_in_g(x1)
[]  =  []
list21_out_g(x1)  =  list21_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U4_G(x1, x2)  =  U4_G(x2)
U5_G(x1, x2)  =  U5_G(x2)
LIST21_IN_G(x1)  =  LIST21_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)

We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LIST21_IN_G(.(T28, T30)) → LIST21_IN_G(T30)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X28))
s2l9_in_ga(s(T16), .(X65, X66)) → U1_ga(T16, X65, X66, s2l9_in_ga(T16, X66))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X65, X66, s2l9_out_ga(T16, X66)) → s2l9_out_ga(s(T16), .(X65, X66))
U3_g(T8, s2l9_out_ga(T8, X28)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, list21_in_g(T23))
list21_in_g([]) → list21_out_g([])
list21_in_g(.(T28, T30)) → U2_g(T28, T30, list21_in_g(T30))
U2_g(T28, T30, list21_out_g(T30)) → list21_out_g(.(T28, T30))
U5_g(T8, list21_out_g(T23)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
list21_in_g(x1)  =  list21_in_g(x1)
[]  =  []
list21_out_g(x1)  =  list21_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
LIST21_IN_G(x1)  =  LIST21_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LIST21_IN_G(.(T28, T30)) → LIST21_IN_G(T30)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LIST21_IN_G(x1)  =  LIST21_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(12) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LIST21_IN_G(.(T30)) → LIST21_IN_G(T30)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • LIST21_IN_G(.(T30)) → LIST21_IN_G(T30)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

S2L9_IN_GA(s(T16), .(X65, X66)) → S2L9_IN_GA(T16, X66)

The TRS R consists of the following rules:

goal1_in_g(s(T8)) → U3_g(T8, s2l9_in_ga(T8, X28))
s2l9_in_ga(s(T16), .(X65, X66)) → U1_ga(T16, X65, X66, s2l9_in_ga(T16, X66))
s2l9_in_ga(0, []) → s2l9_out_ga(0, [])
U1_ga(T16, X65, X66, s2l9_out_ga(T16, X66)) → s2l9_out_ga(s(T16), .(X65, X66))
U3_g(T8, s2l9_out_ga(T8, X28)) → goal1_out_g(s(T8))
goal1_in_g(s(T8)) → U4_g(T8, s2l9_in_ga(T8, T23))
U4_g(T8, s2l9_out_ga(T8, T23)) → U5_g(T8, list21_in_g(T23))
list21_in_g([]) → list21_out_g([])
list21_in_g(.(T28, T30)) → U2_g(T28, T30, list21_in_g(T30))
U2_g(T28, T30, list21_out_g(T30)) → list21_out_g(.(T28, T30))
U5_g(T8, list21_out_g(T23)) → goal1_out_g(s(T8))
goal1_in_g(0) → goal1_out_g(0)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l9_in_ga(x1, x2)  =  s2l9_in_ga(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
0  =  0
s2l9_out_ga(x1, x2)  =  s2l9_out_ga(x2)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
list21_in_g(x1)  =  list21_in_g(x1)
[]  =  []
list21_out_g(x1)  =  list21_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

S2L9_IN_GA(s(T16), .(X65, X66)) → S2L9_IN_GA(T16, X66)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L9_IN_GA(x1, x2)  =  S2L9_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(19) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S2L9_IN_GA(s(T16)) → S2L9_IN_GA(T16)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(21) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • S2L9_IN_GA(s(T16)) → S2L9_IN_GA(T16)
    The graph contains the following edges 1 > 1

(22) YES